Minimal stretch maps between Euclidean triangle?

TL;DR

This paper investigates minimal stretch maps between acute Euclidean triangles, deriving formulas for Lipschitz constants, establishing a symmetric Finsler metric on triangle pairs, and analyzing the geometric properties of this metric space.

Contribution

It introduces a Euclidean analogue of Thurston's asymmetric metric, providing formulas, metric properties, and isometry group characterization for minimal stretch maps between triangles.

Findings

Derived a formula for the Lipschitz constant of minimal stretch maps.

Established that the Lipschitz constant function forms a symmetric Finsler metric.

Characterized geodesics and isometry group of the metric space of triangle pairs.

Abstract

Given two triangles whose angles are all acute, we find a homeomorphism with the smallest Lipschitz constant between them and we give a formula for the Lipschitz constant of this map. We show that on the set of pairs of acute triangles with fixed area, the function which assigns the logarithm of the smallest Lipschitz constant of Lipschitz maps between them is a symmetric metric. We show that this metric is Finsler, we give a necessary and sufficient condition for a path in this metric space to be geodesic and we determine the isometry group of this metric space. This study is motivated by Thurston's asymmetric metric on the Teichm{\"u}ller space of a hyperbolic surface, and the results in this paper constitute an analysis of a basic Euclidean analogue of Thurston's hyperbolic theory. Many interesting questions in the Euclidean setting deserve further attention.